
NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2025 Jan 29, 07:43 -0800
Hello Modris.
Thank you for sharing your analysis! Yes, I agree that this is among the "seemingly trivial problems" in navigation, but it's still interesting and maybe illuminating, right? :) In addition, and perhaps most important of all, there is an ample supply of nonsense on this topic which needs to be rectified. I continue to be amazed by the number of resources that assert that "side error" must be carefully and rigorously zeroed out.
Moving on...
I think we have more to do to get the math right, in part because we place a little too much trust in old Chauvenet! He wasn't perfect. And also his output was a product of its era. It shouldn't be copied down unchanged.
One suggestion, which I strongly recommend: get rid of those damn sin(1") factors! That's an artifact of 19th century trig style, and we should always translate that to something more modern. Otherwise, the equations are left comprehensible only to a small subset of those reading along. It's also a dumb math styling. The factor is not in any meaningful sense a "sine" in the first place. All they wanted from sin(1") was a numerical factor to convert between pure angles (angles as ratios) and sexagesimal angles. Their "sine of one second" should be just pi/(180·3600) or approximately 1/(3438·60) --and normally it should not appear in equations at all.
Is the analysis we usually look at for side error correct in the first place? Let's consider an analysis of side error where the "other object" is the sea horizon. This is by far the most important case for practical sextant use. If I measure the angle between a low altitude star (maybe brilliant Venus, as it's setting this month) and the sea horizon at low height of eye, what is the "cost" to me, the error in my sight, if I have side offset remaining in my sextant optics? I'll suggest as a hypothesis that it's orders of magnitude smaller even than the very small numbers that we have discussed so far. Indeed if the sea horizon is approximated as a great circle (=negligible height of eye), then the side "error" should be exactly zero. Makes sense, yes?
Also, I'll invite everyone following along to consider an analogous situation involving linear measurement. Suppose I have a meter stick with a good scale running along its lower edge from near zero cm out to 100 cm. Suppose also that the zero corner had been chewed off by a busy rabbit. We still have the other corner on the zero end. It's only the zero corner by the scale that is missing, as in my illustration below. Can I measure with this thing? It has a "side offset". What sort of "side error" would I get for various measured lengths? Although it's unlikely in practice, for the sake of argument, let's assume that the width of the stick is 1cm. If I have two "dots" on a board, and I measure the distance between them by placing the scale in contact with one "dot" and the good corner at the zero end in contact with the other "dot", what would I find for the error in the distance between the "dots"?
Next, let's consider a more general sextant case, not a linear "meter stick" analogy and not a case where we are measuring to a sea horizon. But we can limit the analysis in one mathematically useful way. Let's say we're only interested in errors at very small angles. That's fair because we know that the error goes to zero very quickly. So let's just work out the math for small observed angles. Let's assume that the angles (maybe short-distance lunars?) we're measuring are all less than 10°. At that scale we can forget about spherical trigonometry and consider simple plane triangles drawn on a small region of the sky. What do we get for errors in our sextant angles for some given moderate value of side offset? There should be a similarity to the "meter stick" case. There should also be a close match to the spherical triangle solution when it is limited to small angles (what is the value of cot θ when θ is a small angle?).
And finally, where is our experimental data? Let's set up a sextant with a carefully measured side offset. Then let's measure some small angles. Do the results or real-world experiments match mathematical theory?
Just some things to ponder... :)
Frank Reed